1) If you are using the set-and-forget method, the amount of headspace is irrelevant the surface area of the keg (when vertical) is the same whether you have 3 gallons or 5.15 gallons in the keg.
2) If you are shaking then more headspace will carbonate faster since there will be more surface area as you move the keg from vertical. If you connect and disconnect (with 35 or 40 psi) then more headspace will carbonate faster as there is more volume at 40 psi and the pressure will drop more slowly as the beer absorbs the CO2 trapped in the top of the keg.
My preferred method:
Beer should be chilled to 31-32 deg.
Fill keg to just below IN tube.
If beer comes out of the IN tube while filling then pressurize and draw a pint off.
Set regulator to 15lbs, connect the gas in tube, shake on side with gas tube toward the floor.
Shake for 3 minutes.
Put back in 31deg cooler for 1/2 hour to allow foam to settle.
Vent keg to 2-3lbs(ish) Put keg in keezer (still at 32 deg) and hook up to gas.
The beer will be drinkable immediately and will reach optimum carbonation in 2-3 days.